This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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4
votes
3answers
158 views

What type of Hypergeometric series is this?

I am trying to find a closed form for the series $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$ $m$ is a nonzero positive integer, and $b$, $z$...
24
votes
5answers
4k views

What is the mathematical foundation of Control Theory?

There is a question which I'm wondering again and again in recent months. I have taken courses like Elementary Differential Equations, Signals and Systems, Linear Control Systems, General Theory of ...
12
votes
7answers
3k views

Guides/tutorials to learn abstract algebra?

I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not ...
2
votes
0answers
50 views

Inferring a probability distribution from another probability distribution

Let $A$ and $B$ be real-valued random variables, with $f_A$ and $f_B$ their probability density functions. Let's say we can observe the values of $A$ many times and estimate $f_A$ fairly precisely. We ...
2
votes
1answer
273 views

Helpful to review certain calculus topics before first real analysis course?

This is my first time posting, so I apologize in advance if my question is inappropriate here. I wanted to know if it would be beneficial for me to review certain calculus topics before I take my ...
1
vote
0answers
77 views

Suggestions for a reference-level text on optimization theory?

I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
13
votes
1answer
561 views

List videos of interesting courses at the doctoral level.

Many mathematics departments has provided video lessons their courses (usually one semester) that are offered in their doctoral programs in mathematics. Most often these courses total average of 26 ...
1
vote
0answers
31 views

How to write down the maximal subgroups of $GL(9, \mathbb{C})$

I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
4
votes
3answers
118 views

Good source to learn about surface singularities?

I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there. I checked out Miles Reid his lectures on ...
8
votes
1answer
129 views

Amenable group rings embeddable in skew fields

I'm looking for a reference of the following fact: given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent: (1) the group ring $K[G]$ is a domain; (2) $K[G]$ is ...
5
votes
2answers
612 views

What are some reasonable things to prove about the Collatz Conjecture?

I am writing an undergraduate paper on the $3n+1$ problem, and I am looking for some theorems related to the problem that would be reasonable for someone with my mathematical background to prove. I'm ...
3
votes
0answers
358 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
1
vote
3answers
532 views

Introduction to Abstract Harmonic Analysis, reference suggestions?

I'm looking for a good starting book on the subject which only assumes standard undergraduate background. In particular, I need to gain some confidence working with properties of Haar measures, so I ...
9
votes
2answers
326 views

What kind of math topics exist?

What kind of math topics exist? The question says everything I want to know, but for more details: I enjoy studying mathematics but the problem is that I can't find any information with a summary of ...
12
votes
2answers
372 views

Hao Wang's $\mathfrak S$ system/$\Sigma$ system: a “transfinite type” theory that avoids the Goedel's theorems.

Long ago, while I was reading a book ($*$) about the various way to build set theories (Zermelo-Freankel, Von Neumann–Bernays–Gödel, and type theories), I read about a variant of type theory with ...
2
votes
0answers
60 views

Are there modern books (or otherwise) with very strong opinions about the correct way of setting up mathematics?

Euclid had very definite ideas about how to set up mathematics - his method involved axioms, definitions, theorems, and proofs. Similarly, Bourbaki believed in - among other things - proceeding from ...
7
votes
4answers
1k views

What are the prerequisites to Jech's Set theory text?

I'm looking for a book to self-study axiomatic set theory, and heard this was a classic. What are the main prerequisites for this text? My knowledge of set theory isn't too great. Probably the only ...
105
votes
5answers
3k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: $$f(x)=\...
2
votes
1answer
377 views

Problem books in ODE

I'm studying Ordinary differential equations right now in the level of Hartman's book. I've never seen problem books in ODE in this level even if you consider it without solutions. I would like to ...
5
votes
0answers
84 views

Flatness over Jacobson ring

I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where every prime ideal $\mathfrak{p} \subset A$ is an ...
16
votes
2answers
359 views

Basic categories cheat sheet

Has anyone come across a cheat sheet containing basic properties of the most well-known categories (i.e. does it have (co)products, (co)equalizers, (co)limits, etc?)?
2
votes
1answer
477 views

Proof: Mean and Variance of the squared distance of a random walk in n-dimensional space

consider a $x$ step random walk starting from origin in $n$-dimensional space where each step is taken into a random direction and has a distance of 1, i.e., each step is a vector on the $n$-...
3
votes
3answers
314 views

Book recommendation for associative algebras

Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment. ...
6
votes
2answers
214 views

Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ...
3
votes
1answer
117 views

Spectral sequences: equivalence of exact couples and classic (?) method

By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples ...
0
votes
3answers
537 views

Good books on combinatorics

I have a math Ph.D. but my knowledge of combinatorics sucks and I simply don't know how to compute anything more complicated, i.e. what happens when we put restrictions on the allowed configurations ...
2
votes
2answers
635 views

resources to study PDE from

I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
8
votes
3answers
1k views

Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
5
votes
0answers
138 views

A formula by R.L.Graham,AMM(1995)

I see this formula in a book, it comes from R.L.Graham,AMM(1995) : $$\frac{1}{3}=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{7^2}+\frac{1}{54^2}+\frac{1}{112^2}+\frac{1}{640^2}+\frac{1}{4032^2}+\frac{1}{...
3
votes
1answer
139 views

About iterative refinement to the solution of the linear equations

I want to know what is iterative refinement for improving the solution to the linear equations? How they improve solutions and what are the various techniques for the iterative refinements? Any ...
12
votes
1answer
284 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
6
votes
1answer
429 views

An Axiomatic Treatment of Mathematics from First Principles to the Major Subjects?

I'm looking for a book - more likely, books - that could take me from the axioms of mathematical logic up to the major subjects of mathematics, like analysis, algebra, geometry, etc. For example, a ...
2
votes
3answers
132 views

Should I put interpunction after formulas?

I am presently doing my first substantial piece of mathematical writing, hence this, probably somewhat silly, question. How does display-style mathematics interact with punctuation? More precisely,...
2
votes
1answer
203 views

Heighway dragon and twindragon relation

The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1. The ...
12
votes
1answer
561 views

Practical Tips: Mathematical research and discoveries [closed]

How to be when you are working on something innovative? What to do if there is a chance (even the $1\%$) that your work is leading you to something original? For example what have I do if I don't ...
10
votes
1answer
246 views

When the ordinal sum equals the Hessenberg (“natural”) sum

Let $\alpha_1 \geq \ldots \geq \alpha_n$ be ordinal numbers. I am interested in necessary and sufficient conditions for the ordinal sum $\alpha_1 + \ldots + \alpha_n$ to be equal to the Hessenberg ...
1
vote
1answer
35 views

Improving related estimates

There are three underlying quantities $x$, $y$, and $a$, where $x$ and $y$ are vectors, and $a$ is a scalar. They are related by $x = ay$. We get noisy observations, $x_0,y_0$. We want to find $a$, ...
11
votes
2answers
183 views

Origin of well-ordering proof of uniqueness in the FToArithmetic

In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it ...
20
votes
1answer
652 views

Any proof to $\pi^{e}$'s irrationality?

I've searched for this for a while but get nothing... There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
5
votes
1answer
58 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
1
vote
0answers
85 views

Kolmogorov's paper defining Bayesian sufficiency

I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
4
votes
1answer
116 views

The manuscript Summa Logicae (William of Ockham)

The Summa Logicae (Latin, in English it's the Sum of Logic) is a textbook on logic by William of Ockham. There are articles about the Summa Logicae in Wikipedia and in Logicmuseum. It was published ...
2
votes
3answers
633 views

Thermodynamics for math majors

I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. ...
0
votes
1answer
164 views

A matrix has a real logarithm if it has a positive spectrum.

The title is a proposition I read in my notes that's left with no proof. Where can I read one?
5
votes
1answer
222 views

Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$

My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$. I'm a bit ...
3
votes
1answer
194 views

Figure $\infty$ is immersion of circle

Where can I find prove of: Figure $\infty$ is immersion of circle. More thanks for a prove or a function between these manifolds.
5
votes
2answers
427 views

Applications of design theory

I have recently started reading up on design theory, with the ultimate purpose of doing some original research in that area. I understand the mathematics fairly well, but am not understanding the ...
6
votes
3answers
2k views

What's a good book for a beginner in high school math competitions?

Also, I want to make it clear: Beginner. I'm getting really frustrated trying to study for math competitions: On the one hand, there are books teaching the high school curriculum, but that's it. I ...
1
vote
0answers
68 views

Are there books on algorithms for architecture?

I need books on algorithms for organic/nonlinear and linear architect anyone recommend a book?
5
votes
0answers
95 views

Solution to $\Delta_g u = \delta-1$ on a 2-sphere.

Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...